Prob Sheet 1: First Order Ordinary Differential Equations
Module F13YT2
1.
Integration revision. Find y(x), given that
dy
dy
ex
(i)
; (ii)
=
= x2 sin x.
x
dx
3 + 6e
dx
2.
Name the type and find the general solution for each of the following first order equations:
dy
dy
x2
dy
(i)
=
; (ii)
+ 3y = x + e−2x ; (iii) x
= x cos(2x) − y;
dx
y
dx
dx
y 2 + 2xy
dy
dy
=
= 0;
(iv)
;
(v) xy 2 − x + (x2 y + y)
dx
x2
dx
dy
(vi) (xey + 2y)
+ ey = 0;
dx
dy
+ 1 − y sin x + e2y = 0
(vii) (cos x + 2xe2y )
dx
dy
(viii)
+ y = 2 y3
dt
3.
Show that each of the following equations has two distinct solutions.
dy
dy
(ii) y
(i)
= y 1/2 ; y(0) = 0;
= x; y(0) = 0.
dx
dx
Explain, in each case, why the hypotheses of Picard’s Theorem are not satisfied.
dy
+ y 2 = 1. Hence sketch the solutions satisfying
dx
(i) y(0) = 1.1; (ii) y(0) = 0; (iii) y(0) = −1.1. Check your answer using Maple.
4.
Find all constant solutions of
dy
Consider the equation
= y 2 − x. Determine the points in the xy-plane at which
dx
dy
dy
dy
(i)
= 0; (ii)
> 0; and (iii)
< 0. Hence sketch the solution satisfying y(1) = 1.
dx
dx
dx
Check your answer using Maple.
5.
6. Consider the following first order differential equation
dy
= y − x.
dx
(∗)
(a) Determine the points in the xy-plane at which
dy
dy
dy
= 0; (ii)
> 0; and (iii)
< 0. Hence, sketch the direction field of (∗).
(i)
dx
dx
dx
(b) Sketch the solution of (∗) satisfying the initial condition y(0) = 12 in the direction field you
have made for (a) (there is no need to solve the initial value problem explicitly).